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The Complexity of Equivalence and Isomorphism of Systems of Equations over Finite Groups
Linköping University, Department of Computer and Information Science, TCSLAB - Theoretical Computer Science Laboratory. Linköping University, The Institute of Technology.
2005 (English)In: Theoretical Computer Science, ISSN 0304-3975, Vol. 345, no 2-3, 406-424 p.Article in journal (Refereed) Published
Abstract [en]

We study the computational complexity of the isomorphism and equivalence problems on systems of equations over a fixed finite group. We show that the equivalence problem is in P if the group is Abelian, and coNP-complete if the group is non-Abelian. We prove that if the group is non-Abelian, then the problem of deciding whether two systems of equations over the group are isomorphic is coNP-hard. If the group is Abelian, then the isomorphism problem is GRAPH ISOMORPHISM-hard. Moreover, if we impose the restriction that all equations are of bounded length, then we prove that the isomorphism problem for systems of equations over finite Abelian groups is GRAPH ISOMORPHISM-complete. Finally, we prove that the problem of counting the number of isomorphisms of systems of equations is no harder than deciding whether there exist any isomorphisms at all.

Place, publisher, year, edition, pages
2005. Vol. 345, no 2-3, 406-424 p.
Keyword [en]
Graph isomorphism; Computational complexity; Systems of equations; Finite groups
National Category
Engineering and Technology
URN: urn:nbn:se:liu:diva-14452DOI: 10.1016/j.tcs.2005.07.018OAI: diva2:23524
Available from: 2007-05-03 Created: 2007-05-03 Last updated: 2009-05-28
In thesis
1. Complexity Dichotomies for CSP-related Problems
Open this publication in new window or tab >>Complexity Dichotomies for CSP-related Problems
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Ladner’s theorem states that if PNP, then there are problems in NP that are neither in P nor NP-complete. Csp(Γ) is a class of problems containing many well-studied combinatorial problems in NP. Csp(Γ) problems are of the form: given a set of variables constrained by a set of constraints from the set of allowed constraints Γ, is there an assignment to the variables satisfying all constraints? A famous, and in the light of Ladner’s theorem, surprising conjecture states that there is a complexity dichotomy for Csp(Γ); that is, for any fixed finite Γ, the Csp(Γ) problem is either in P or NP-complete.

In this thesis we focus on problems expressible in the Csp(Γ) framework with different computational goals, such as: counting the number of solutions, deciding whether two sets of constraints have the same set of solutions, deciding whether all minimal solutions of a set of constraints satisfies an additional constraint etc. By doing so, we capture a host of problems ranging from fundamental problems in nonmonotonic logics, such as abduction and circumscription, to problems regarding the equivalence of systems of linear equations. For several of these classes of problem, we are able to give complete complexity classifications and rule out the possibility of problems of intermediate complexity. For example, we prove that the inference problem in propositional variable circumscription, parameterized by the set of allowed constraints Γ, is either in P, coNP-complete, or ΠP/2-complete. As a by-product of these classifications, new tractable cases and hardness results for well-studied problems are discovered.

The techniques we use to obtain these complexity classifications are to a large extent based on connections between algebraic clone theory and the complexity of Csp(Γ). We are able to extend these powerful algebraic techniques to several of the problems studied in this thesis. Hence, this thesis also contributes to the understanding of when these algebraic techniques are applicable and not.

Place, publisher, year, edition, pages
Institutionen för datavetenskap, 2007. 36 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1091
Complexity, Constraint Satisfaction Problem, System of Equations, Nonmonotonic Logic, Circumscription, Abduction, Isomorphism
National Category
Computer Science
urn:nbn:se:liu:diva-8822 (URN)978–91–85715–20–6 (ISBN)
Public defence
2007-06-01, Visionen, Hus B, Campus Valla, Linköping University, Linköping, 13:15 (English)
Available from: 2007-05-03 Created: 2007-05-03 Last updated: 2009-09-08Bibliographically approved

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